cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A271959 G.f. A(x) satisfies: A(x)^2 = A( x^2 + 2*A(x)^3 ), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 3, 11, 46, 206, 968, 4706, 23475, 119473, 617911, 3238299, 17159235, 91778276, 494844531, 2686731634, 14676812392, 80608719674, 444853616294, 2465582096210, 13718412623437, 76596933043436, 429046752044241, 2410260414325754, 13576401507921773, 76660997896449976, 433862051647272420, 2460614616704514931, 13982516582761530427, 79601022639916142384
Offset: 1

Views

Author

Paul D. Hanna, Apr 18 2016

Keywords

Comments

Compare the g.f. to the related identity:
C(x)^2 = C( x^2 - 2*C(x)^3 ), where C(x) = x - C(x)^2.

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 206*x^6 + 968*x^7 + 4706*x^8 + 23475*x^9 + 119473*x^10 + 617911*x^11 + 3238299*x^12 +...
where A(x)^2 = A( x^2 + 2*A(x)^3 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 28*x^5 + 123*x^6 + 570*x^7 + 2745*x^8 + 13596*x^9 + 68818*x^10 + 354380*x^11 + 1850642*x^12 + 9777476*x^13 + 52166536*x^14 +...
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 52*x^6 + 240*x^7 + 1152*x^8 + 5694*x^9 + 28776*x^10 + 148008*x^11 + 772208*x^12 + 4076736*x^13 + 21737472*x^14 +...
Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then
B(x) = x - x^2 - x^3 - x^4 - 2*x^5 - 3*x^6 - 6*x^7 - 11*x^8 - 23*x^9 - 46*x^10 - 98*x^11 - 207*x^12 - 451*x^13 - 983*x^14 +...+ -A001190(n)*x^(n+1) +...
such that B(x) = x - x*G(x), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)).
SPECIFIC VALUES.
A(1/6) = 0.268639354433758631443638721883026384052966634356654...
where A(1/6)^2 = A( 1/36 + 2*A(1/6)^3 ).
A(1/7) = 0.185430467497916613031797200968643881842140126550450...
where A(1/7)^2 = A( 1/49 + 2*A(1/7)^3 ).
A(1/8) = 0.152603770337160474296825145654940422398214899394916...
where A(1/8)^2 = A( 1/64 + 2*A(1/8)^3 ).
A(1/10) = 0.11495842465953100301539082058016718103413837897733...
where A(1/10)^2 = A( 1/100 + 2*A(1/10)^3 ).

Links

Crossrefs

Cf. A001190, A271960.

Programs

  • PARI
    {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, X^2 + 2*A^3)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A(x - x*G(x)) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of the Wedderburn-Etherington numbers (A001190).
G.f. A(x) satisfies: A( sqrt(x*F(x)) ) = F(x), where F(x) is the g.f. of A271960 and F(x)^2 = F( (x + 2*F(x)^2)^2 ). - Paul D. Hanna, Aug 09 2024
a(n) ~ c * d^n / n^(3/2), where d = 5.99301788836820936729... and c = 0.060358293047581601577... . - Vaclav Kotesovec, May 03 2016

A376224 G.f. A(x) satisfies A( (x + 3*A(x)^2)^3 ) = A(x)^3.

Original entry on oeis.org

1, 3, 18, 136, 1152, 10458, 99473, 978480, 9872181, 101598389, 1062382809, 11255336235, 120555453344, 1303305334704, 14202627395202, 155847144409224, 1720542786453765, 19096869133735155, 212977164179543266, 2385405242723601582, 26820428322385799784, 302611771988083401990
Offset: 1

Views

Author

Paul D. Hanna, Oct 13 2024

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 18*x^3 + 136*x^4 + 1152*x^5 + 10458*x^6 + 99473*x^7 + 978480*x^8 + 9872181*x^9 + 101598389*x^10 + ...
where A( (x + 3*A(x)^2)^3 ) = A(x)^3.
RELATED SERIES.
A(x)^3 = x^3 + 9*x^4 + 81*x^5 + 759*x^6 + 7362*x^7 + 73386*x^8 + 747567*x^9 + 7749720*x^10 + 81500094*x^11 + 867420469*x^12 + ...
( x^2*A(x) )^(1/3) = x + x^2 + 5*x^3 + 35*x^4 + 284*x^5 + 2508*x^6 + 23401*x^7 + 226950*x^8 + 2265015*x^9 + 23110418*x^10 + ...
Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then
B(x) = x - 3*x^2 - x^4 - x^7 - 2*x^10 - 4*x^13 - 9*x^16 - 22*x^19 - 55*x^22 - 142*x^25 - 376*x^28 - ... + -A352702(n)*x^(3*n+4) + ...
where B(x) = x*(1 - x*G(x))^3 and B(x) = x - 3*x^2 - x^4*G(x^3), where G(x) is the g.f. of A352702 and begins:
G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 55*x^6 + 142*x^7 + 376*x^8 + 1011*x^9 + 2758*x^10 + ...

Links

Crossrefs

Cf. A352702, A271960.

Programs

  • PARI
    {a(n) = my(A = x+x^2); for(m=1, n, A = truncate(A) + x^2*O(x^m); A = subst(A, x, (x + 3*A^2)^3 )^(1/3) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( (x + 3*A(x)^2)^3 ).
(2) x = A( x*(1 - x*G(x))^3 ), where G(x) is the g.f. of A352702.
(3) x = A( x - 3*x^2 - x^4*G(x^3) ), where G(x) is the g.f. of A352702.
a(n) ~ c * d^n / n^(3/2), where d = 12.108643088449238597222614925208058784697264797459219306522454237465345359... and c = 0.0455800108980650629231383349217685291247499776153219609599892816651... - Vaclav Kotesovec, Oct 14 2024

A378247 G.f. A(x) satisfies A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).

Original entry on oeis.org

1, 1, 3, 10, 39, 161, 699, 3135, 14427, 67716, 322959, 1560585, 7624007, 37593476, 186856061, 935214523, 4709265692, 23841104525, 121275951719, 619558165489, 3177346503440, 16351749778167, 84419824808865, 437105510426235, 2269266695980449, 11810014285000263, 61602685079710638
Offset: 1

Views

Author

Paul D. Hanna, Nov 20 2024

Keywords

Comments

Compare to C(x)^2 = C( x^2 + 2*x*C(x)^2 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 161*x^6 + 699*x^7 + 3135*x^8 + 14427*x^9 + 67716*x^10 + 322959*x^11 + 1560585*x^12 + ...
where A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).
RELATED SERIES.
Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
A( x - x^2 - x*G(x^2) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + 2179*x^14 + ... + A001190(n)*x^n + ...
Let B(x) be the series reversion of g.f. A(x) so that B(A(x)) = x, then
B(x) = x - x^2 - x^3 - x^5 - x^7 - 2*x^9 - 3*x^11 - 6*x^13 - 11*x^15 - 23*x^17 - 46*x^19 - 98*x^21 - 207*x^23 + ...
where B(x) = x - x^2 - x*G(x^2).
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.1804894059505127351310871071614416167035910065610113327...
  where 1/9 = A( t^2 + 2*t/9 + 2/81 ).
A(t) = 1/4 at t = 0.1708289565101545485579649480920097855916395263217351536...
  where 1/16 = A( t^2 + t/8 + 1/128 ).
A(t) = 1/5 at t = 0.1516661092515691718015998101146470241027491658579501286...
  where 1/25 = A( t^2 + 2*t/25 + 2/625 ).
A(t) = 1/6 at t = 0.1341268789797555579297424694390747929782019601987848246...
  where 1/36 = A( t^2 + t/18 + 1/648 ).
A(1/6) = 0.2368314953172156547771056118501694080205525703518284958...
A(1/7) = 0.1824082884402163049324182135107985537409785918465705698...
A(1/8) = 0.1515179821748020682616541846638756124979071552818869937...
A(1/9) = 0.1303577455916869424988611259176631850931169441135101392...
A(1/10) = 0.1146797533131163787803333792504789207692884367435306666...

Links

Crossrefs

Cf. A001190, A374567, A271959, A271960, A374566.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2 + 2*x*Ax^2 + 2*Ax^4) - Ax^2, #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas in which G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.
(1) A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).
(2) x = A( x - x^2 - x*G(x^2) ).
(3) x = A( x + x^2 - 2*x*G(x) + x*G(x)^2 ).
(4) x = A( x*sqrt(1 - 2*x^2 - G(x^4)) - x^2 ).
(5) x^2 = A( x^2*((1 - G(x))^2 + 2*x)^2 + x^4 ).
(6) G(A(x)) = 1 - sqrt(x/A(x) - A(x)).
Showing 1-3 of 3 results.

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